Groupes de Picard et problèmes de Skolem. I. (Picard groups and Skolem problems. I). (French) Zbl 0704.14014

Moret-Bailly, Laurent

Ann. Sci. Éc. Norm. Supér. (4) 22, No. 2, 161-179 (1989).

A problem of Skolem asks about the existence of a solution of a system of diophantine equations over a ring R of algebraic integers whose coordinates belong to a finite extension of R. In the scheme-theoretical language this is a question about the existence of a multi-section of a morphism of finite type \(f:X\to Spec(R)\), where R is a Dedekind ring.
The main result of this paper asserts that the answer is positive if the following conditions are satisfied:
(i) R is an excellent ring whose residue fields are algebraic over a finite field;
(ii) for any finite extension \(K'\) of the field of fractions K of R the normalization \(R'\) of R in \(K'\) has torsion Picard group;
(iii) there exists \(K'\) as above such that one of the irreducible components of the base change \(X\otimes_ RR'\) is mapped surjectively to \(Spec(R').\)
This result is a generalization of a theorem of R. S. Rumely [J. Reine Angew. Math. 368, 127-133 (1986; Zbl 0581.14014)], who assumed that R is a ring of integers in an algebraic number field and f is surjective. The proof is geometric and does not use the theory of capacity of Rumely.
[See also the following review.]

Reviewer: I.V.Dolgachev

MSC:

14G25	Global ground fields in algebraic geometry
13F05	Dedekind, Prüfer, Krull and Mori rings and their generalizations
11R04	Algebraic numbers; rings of algebraic integers
11D41	Higher degree equations; Fermat’s equation

Keywords:

Skolem problems; finite extension ring of algebraic integers; existence of a solution of a system of diophantine equations

Citations:

Zbl 0704.14015; Zbl 0581.14014

References:

[1]	M. ARTIN , Some Numerical Criteria for Contractibility of Curves on Algebraic Surfaces (Amer. J. Math., vol. 84, 1962 , p. 485-496). MR 26 #3704 | Zbl 0105.14404 · Zbl 0105.14404 · doi:10.2307/2372985
[2]	D. CANTOR et P. ROQUETTE , On Diaphantine Equations over the Ring of All Algebraic Integers (J. of Number Theory, vol. 18, 1984 , p. 1-26). MR 85j:11036 | Zbl 0538.12014 · Zbl 0538.12014 · doi:10.1016/0022-314X(84)90038-6
[3]	P. DELIGNE et D. MUMFORD , The Irreducibility of the Space of Curves of Given Genus (Pub. Math. I.H.E.S., n^\circ 36). Numdam | Zbl 0181.48803 · Zbl 0181.48803 · doi:10.1007/BF02684599
[4]	W. FULTON , Intersection Theory , Springer Ergebnisse, vol. 2. · Zbl 0541.14005
[5]	J.-P. JOUANOLOU , Théorèmes de Bertini et applications (Progress in Math., vol. 42, Birkhöuser). MR 86b:13007 | Zbl 0519.14002 · Zbl 0519.14002
[6]	D. KNUTSON , Algebraic Spaces (Lecture Notes in Math., vol. 203, Springer). MR 46 #1791 | Zbl 0221.14001 · Zbl 0221.14001 · doi:10.1007/BFb0059750
[7]	L. MORET-BAILLY , Points entiers des variétés arithmétiques [Séminaire de théorie des nombres de Paris (Progress in Math., Birkhöuser)]. Zbl 0644.14008 · Zbl 0644.14008
[8]	L. MORET-BAILLY , Groupes de Picard et problèmes de Skolem II [Ann. sci. Ec. Norm. Sup. (4e série, t. 22, 1989 , p. 181-194)]. Numdam | MR 90i:11065 | Zbl 0704.14015 · Zbl 0704.14015
[9]	L. MORET-BAILLY , Problèmes de Skolem sur les champs algébriques (en préparation). · Zbl 1106.11022
[10]	D. MUMFORD , The Topology of Normal Singularities of an Algebraic Surface and a Criterion for Simplicity (Pub. Math. I.H.E.S., n^\circ 9, 1961 ). Numdam | MR 27 #3643 | Zbl 0108.16801 · Zbl 0108.16801 · doi:10.1007/BF02698717
[11]	R. RUMELY , Capacity Theory on Algebraic Curves (à paraître). · Zbl 0679.14012
[12]	R. RUMELY , Arithmetic Over the Ring of all Algebraic Integers (J. reine u. angew. Math., 368, 1986 , p. 127-133). MR 87i:11041 | Zbl 0581.14014 · Zbl 0581.14014 · doi:10.1515/crll.1986.368.127
[13]	T. SKOLEM , Lösung gewisser Gleichungen in ganzen algebraischen Zahlen, insbesondere in Einheiten (Skrifter Norske Videnskaps-Akademi i Oslo, Mat. Naturv. Kl. 10, 1934 ). Zbl 0011.19701 | JFM 61.0175.06 · Zbl 0011.19701
[14]	L. SZPIRO et coll. Séminaire sur les pinceaux de courbes de genre au moins deux (Astérisque, vol. 86). · Zbl 0463.00009